Exponential sums with a large second derivative (Q2710120)

Exponential sums of the form \(S=\sum_{M \leq m \leq M_2}e(TF(m/M))\) are considered, where \(M\leq M_2 \leq 2M\) and \(F(x)\) is a real differentiable function for \(1\leq x \leq 2\). The nature of this sum depends on the size of the parameter \(\alpha =\log M/\log T\). The case where \(\alpha \) is close to \(1/2\) is relevant in the application to the ``Lindelöf problem'' for Riemann's zeta-function, and in this case the well-known Bombieri-Iwaniec method was successful. In that method, the sum \(S\) is divided into shorter sums (``Farey arcs'') according to a system of Farey fractions, and one has to deal with ``resonances'' between different Farey arcs. The present paper is concerned with the range \(2/5\leq \alpha \leq 3/7\) utilizing a peculiarity of the resonance problem in this case, and a new estimate for \(S\) is obtained if \(\alpha \) lies in a certain range. For instance, the present method is said to be the most powerful known for \(\alpha = 7/17 = 0.4118... \).NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].